Quantum structural fluxion in superconducting lanthanum polyhydride

The discovery of 250-kelvin superconducting lanthanum polyhydride under high pressure marked a significant advance toward the realization of a room‐temperature superconductor. X-ray diffraction (XRD) studies reveal a nonstoichiometric LaH9.6 or LaH10±δ polyhydride responsible for the superconductivity, which in the literature is commonly treated as LaH10 without accounting for stoichiometric defects. Here, we discover significant nuclear quantum effects (NQE) in this polyhydride, and demonstrate that a minor amount of stoichiometric defects will cause quantum proton diffusion in the otherwise rigid lanthanum lattice in the ground state. The diffusion coefficient reaches ~10−7 cm2/s in LaH9.63 at 150 gigapascals and 240 kelvin, approaching the upper bound value of interstitial hydrides at comparable temperatures. A puzzling phenomenon observed in previous experiments, the positive pressure dependence of the superconducting critical temperature Tc below 150 gigapascals, is explained by a modulation of the electronic structure due to a premature distortion of the hydrogen lattice in this quantum fluxional structure upon decompression, and resulting changes of the electron-phonon coupling. This finding suggests the coexistence of the quantum proton fluxion and hydrogen-induced superconductivity in this lanthanum polyhydride, and leads to an understanding of the structural nature and superconductivity of nonstoichiomectric hydrogen-rich materials.

The simulated XRD patterns of the static VT and VC structures of LaH9.97 are almost identical to that of the fcc-LaH10 at 150 GPa, whereas the H-H separations in the VT and VC structures of LaH9.97 distribute more broadly than that in fcc-LaH10, indicating that the lattice distortion spreads out away from a vacancy site in the clathrate H framework of LaH9.97. The 'a.u.' is the abbreviation of arbitrary unit, throughout the article. The dΩ/dp (with Ω being or ̅ 2 ) for estimating the dTc/dp in quantum fluxional LaH9.63 is estimated based on the ( ) of LaH9.63 obtained by Fourier transforming the velocity autocorrelation functions in the CMD simulations at 240 K in combination with coupling functions ( ) 2 approximated by that of quantum , we derived dΩ/dp and analyzed the role played by several parameters in the pressure trend of Tc using AD equations, with the Coulomb coupling constant μ* set to 0.1. The resulting pressure trends of Tc for cubic LaH10 and CaH6 are in good agreement with the literature (Supplementary Figure 1).
The analysis demonstrates that decreases more than two times faster than increases upon compression, which leads to a negative sign of the dλ/dp slope. This suggests that d /dp dominates the pressure trend of λ.
λ has a monotonously negative pressure dependence, but decreases less significantly than increases.
However, due to the nonlinear dependence of Tc on λ, either through the well-known exponential function or the 'strong-coupling correction' and 'shape correction' factors (see details in page 14), dλ/dp plays a dominant role in determining the dTc/dp slope.
Our results for LaH9.63 are also shown in order to enable a direct comparison. Our d / dp is comparable to those of cubic LaH10 (fcc) and CaH6 (bcc), but d /dp decreases significantly at lower pressures, which reverses the sign of the dTc/dp slope through a considerable reduction of dλ/dp below 163 GPa. We further estimated the limit of stability of the +dTc/dp slope against variations of and d /dp: I) at a fixed d /dp of 0.01235 eV 3 /megabar, decreasing (increasing) by 50 % results in a decrease (an increase) of dTc/dp by 59% (41%), but does not change the positive sign; II) at a fixed of 0.04352 eV 3 at 176 GPa, an about 127 % decrease of the d /dp slope is needed to invert the sign of dTc/dp. Literature data generally report a positive d /dp slope for cubic hydrides covering a range of H/M ratios (M= La and Ca) from 10 to 6. Therefore, LaH9.63 is very unlikely to exhibit a -d /dp slop in qualitative contradiction with most other hydrides. The analysis indicates that the +dTc/dp slope is robust to errors induced by the approximation of and d /dp in LaH9.63 by those in quantum fcc-LaH10. listed here for illustration. Furthermore, we compare the pressure trends of Tc calculated for various of ( ) 2 (i.e. 0, 0.01 and 0.05). It is found that the trend is determined by the overall shape of ( ) 2 . Due to the inexistence of a method that can directly calculate the 2 ( ) of a quantum fluxional structure, we estimated frequency moments Ω of 2 ( ) in LaH9.63 based on the calculated ( ) of LaH9.63 and the ( ) 2 approximated by that of quantum fcc-LaH10 12 . To evaluate the potential errors associated with the approximation, we calculated the Tc using reshaped 2 ( ) obtained by artificial scaling of . Empirically, EPC weights the phonon spectrum to lower frequencies in high-Tc hydrides. This weighting effect changes smoothly with pressure, as observed from the shape of ( ) 2 in quantum fcc-LaH10 (Supplementary Figure   7). Considering this, and that quantum LaH9.63 and fcc-LaH10 exhibit a similar phonon hardening trend with increasing pressure as well as similar shapes of ( ), our results suggest that the pressure trend of Tc is robust to variations of Ω. This indicates that it is practicable to estimate qualitatively the sign of dTc/dp slop in LaH9.63 under this approximation.

Molecular dynamics calculations
Quantum nuclear dynamics were studied using path-integral molecular dynamics (PIMD) and centroid molecular dynamics (CMD), with the massive Nosé-Hoover chain (NHC) thermostats on NVT and NVE ensembles (N-number of particles; V-volume; T-temperature, and E-energy), respectively, as implemented in the PIMD code 17 . NpT-PIMD (ppressure) simulations were performed to estimate the quantum pressure-volume relations. Beads number was set to 16 for the cases without a specification, and time step was set to 0.5 and 0.05 fs for PIMD and CMD, respectively. A short PIMD simulation was always carried out on the initial structure to prepare the pre-equilibrium state in order to promote
The quantum pressure-volume relation of fcc-LaH10 and fcc-LaH9 at 300 K was estimated from the volume averaged

Vacancy formation enthalpy, H f
We carried out a full variable-cell optimization (using ISIF=3 in VASP) of fcc-LaH10 7, 8 (in a 2×2×2 supercell of the conventional unit cell containing 352 atoms) and B2/n-H2 (in the conventional unit cell of 48 atoms) 19 at pressures between 120 and 220 GPa with an interval of 5 GPa. We employed k-mesh grids of sizes 3×3×3 and 15×9×9, respectively, to carry out the required DFT calculations.
The VT and VC LaH9.97 vacancy structures were then constructed by removing the appropriate H atom from the relaxed fcc-LaH10 structure at each pressure, as described in the main text. Variable-cell optimization of these structures induced proton diffusion, resulting in a distortion of the structures. In order to isolate the effect of the vacancies, therefore, we only relaxed the atomic positions, keeping the cell shape and volume fixed (ISIF=2 in VASP). We used a k-mesh grid equivalent to that for fcc-LaH10. A cut-off energy of 325 eV and a convergence criterion for the total energy of 3×10 −7 eV/atom were employed in all calculations.
Using the resulting V-E relationship of the relaxed structures, the pressure-enthalpy curves were then calculated using The calculated pressure-volume curve gives a pressure shift of ~5 GPa for LaH9.97 with respect to the experimental measurements on fcc-LaH9.6 at 120-220 GPa, which does not affect our discussions.
It should be noted that H f is calculated classically in the static lattice approximation on the Born-Oppenheimer energy surface, which differs from the configurational energy surface of the quantum crystal (e.g., fcc-LaH10 in Ref. 12) or the free energy surface of the classical or quantum crystal at finite temperature. The stability and pressure boundary of the 12／17 vacancy structure may be affected by more complex physical effects existing in the superconducting samples, for instance nuclear quantum effects (including zero-point motion of nuclei and proton tunnelling), or temperature effects.
Indeed, temperature effects only enhance the formation of vacancies. In this sense, the conclusion from the H f curves shown in Fig.1a of the main text, namely that vacancy formation is favored below 158 GPa in the classical crystal picture, while being based on a model calculation subject to many approximations, indicates that lanthanum superhydride should be rather prone to vacancy formation.

The configurational distance, ξ
The crystal fingerprint technique utilized in this study is based on Gaussian overlap matrices, which represent the local environments of all atoms in a unit cell and can efficiently determine configurational distances ξ between crystalline structures satisfying the mathematical requirements of a metric 21

Electronic density of states at the Fermi level, ( )
The ( ) is of central importance to the understanding of conventional superconductivity for hydrogen-rich materials.
It is quite general that the pressure dependence of ( ) directly correlates to the pressure trend of the superconducting   μ* is the Coulomb coupling constant, which is set to a typical value of 0.1 in this wok.

Structural information of static LaH10-.
The simulation cells of fcc-LaH9 (320 atoms) and fcc-LaH10 (352 atoms) are initialized by a 2×2×2 extension of the unit cell of F-43m-LaH9 13 and Fm-3m-LaH10 12 , respectively. We initialize the LaH9.63 structure (340 atoms) by adding randomly 20 H atoms to the simulation cell of fcc-LaH9, constraining neighboring vacancies to be no less than 4.4 Å apart to ensure an approximately uniform vacancy distribution, as shown in Supplementary Figure 11. A PIMD simulation (2000 steps) was always carried out on the initial structure to prepare the pre-equilibrium state for the CMD simulation. Vacancy diffusion generally results in a random distribution of vacancies on the 8c and 32f Wyckoff position in Fm-3m-LaH10 within several hundred PIMD steps. The static P1-LaH9.63, mimicking the quantum fluxional LaH9.63, is built by scaling the lattice parameter a of a QFS sampled from the CMD simulation of LaH9.63 at 176 GPa. The initial vacancy structure of LaH9.63 and static P1-LaH9.63 at 150 GPa can be found at the linkhttps://doi.org/10.6084/m9.figshare.20484420.v1. Figure 11. Initial vacancy structure of LaH9.63.